Finite Group Databasebeta

← #14511704

Subgroups — #14511704

Order 1920.

252 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 37 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1SL(2,𝔽3)×D20C211yesmaximal, characteristic
2C5×#5386C211yesmaximal, characteristic, complemented
3SL(2,𝔽3)⋊#168C211yesmaximal, characteristic, complemented
4#10601920—31nomaximal, Hall
5C2×D5×SL(2,𝔽3)—21no—
6C4×C5×SL(2,𝔽3)C2211yescharacteristic
7C2×#6655—41nomaximal
8C5×#859—21no—
9#73087—51nomaximal, Hall
10Q8×D20D311yescharacteristic
11C5×#475—31no—
12#65372—31no—
13#65368—31no—
14Q8⋊#168—31no—
15#64703—31no—
16#64702—31no—
17C2×C5×Dic6—41no—
18C2×C3×D20—41no—
19C2×C5×SL(2,𝔽3)D411yescharacteristic
20D5×SL(2,𝔽3)—41no—
21C2×C3⋊#168—41no—
22#6655—44no—
23D4×SL(2,𝔽3)—51no—
24#5386D511yescharacteristic, complemented
25SL(2,𝔽3)⋊C8—51no—
26C2×Q8×D5—21no—
27C5×C4⋊Q8—31no—
28C4×C5×Q8C2×D311yescharacteristic
29C2×#688—61no—
30#4043—61no—
31C4×D20—31no—
32#4015—31no—
33C5×#106—61no—
34C5×C4⋊C8—31no—
35C5×#103—61no—
36#3967—61no—
37#3940—62no—
38C4⋊#168—31no—
39C4×#168—31no—
40#1651—151noSylow
41C22×C3×D5—81no—
42C2×C3×C4×C5—41no—
43C2×C5×Dic3—81no—
44C5×Dic6—42no—
45C5×Dic6—81no—
46C3×D20—42no—
47C3×D20—81no—
48C5×SL(2,𝔽3)SD1611yescharacteristic, complemented
49C3⋊#168—42no—
50C22×SL(2,𝔽3)—101no—
51C2×#214—201no—
52C4×SL(2,𝔽3)C2×D511yescharacteristic
53#859—21no—
54C2×C5×Q8—61no—
55C2×C5×Q8#6711yescharacteristic
56Q8×D5—41no—
57Q8×D5—121no—
58C2×D20S411yescharacteristic
59C2×C4×D5—61no—
60C2×Dic10—61no—
61C2×C5×C8—61no—
62C5×C4⋊C4—31no—
63C5×C4⋊C4—61no—
64C5×C4⋊C4—61no—
65C42×C5—31no—
66#688—122no—
67#685—61no—
68C4⋊Dic5—32no—
69C4⋊Dic5—31no—
70C2×#168—61no—
71C2×#168—61no—
72D4×Q8—51no—
73#498—151no—
74#475—31no—
75#461—151no—
76#332—151no—
77Q8⋊C8—151no—
78D4⋊C8—151no—
79C22×C3×C5—41no—
80C2×C3×D5—82no—
81C2×C3×D5—161no—
82C2×C3×D5—161no—
83C3×C4×C5—41no—
84C3×C4×C5—41no—
85C5×Dic3—82no—
86C2×C3×D4—201no—
87C2×Dic6—41no—
88C2×SL(2,𝔽3)—201no—
89C2×SL(2,𝔽3)C22⋊D511yescharacteristic
90#214—204no—
91C2×C3⋊C8—201no—
92C22×D5—21no—
93C5×Q8—31no—
94C5×Q8—121no—
95C5×Q8—121no—
96C5×Q8#21411yescharacteristic, complemented
97C2×C4×C5—31no—
98C2×C4×C5—31no—
99C2×C4×C5—121no—
100C2×C4×C5C2×S411yescharacteristic
101C2×Dic5—61no—
102D20—21no—
103D20GL(2,𝔽3)12yes—
104C4×D5—121no—
105C4×D5—121no—
106Dic10—121no—
107Dic10—121no—
108C5×C8—121no—
109#168—62no—
110#168—121no—
111C22×Q8—101no—
112C2×SD16—301no—
113C4⋊Q8—31no—
114C4⋊Q8—151no—
115C22⋊Q8—301no—
116C4×Q8D3×D511yescharacteristic
117C4×D4—151no—
118#106—61no—
119#106—302no—
120C4⋊C8—31no—
121C4⋊C8—151no—
122#103—61no—
123#103—301no—
124C4×C8—151no—
125C2×C3×C5—41no—
126C2×C3×C5—41no—
127C2×C3×C5—41no—
128C3×D5—162no—
129C23×C3—401no—
130C3×D4—202no—
131C3×D4—401no—
132C2×C3×C4—41no—
133C2×Dic3—81no—
134Dic6—42no—
135Dic6—81no—
136SL(2,𝔽3)#68811yescharacteristic, complemented
137C3⋊C8—202no—
138C22×C5GL(2,ℤ/4)11yescharacteristic
139C2×D5—22no—
140C2×D5—41no—
141C2×D5—41no—
142C4×C5—31no—
143C4×C5—31no—
144C4×C5—61no—
145C4×C5—241no—
146C4×C5C2×GL(2,𝔽3)11yescharacteristic
147C4×C5#98311yescharacteristic
148Dic5—121no—
149C2×Q8—61no—
150C2×Q8—201no—
151C2×Q8—301no—
152C2×Q8—601no—
153C2×Q8#120111yescharacteristic
154C2×D4—51no—
155C22×C4—301no—
156SD16—602no—
157C2×C8—61no—
158C2×C8—301no—
159C2×C8—301no—
160C4⋊C4—31no—
161C4⋊C4—61no—
162C4⋊C4—61no—
163C4⋊C4—152no—
164C4⋊C4—151no—
165C22⋊C4—301no—
166C42—31no—
167C3×C5—41noHall
168C22×C3—41no—
169C22×C3—402no—
170C22×C3—801no—
171C22×C3—801no—
172C3×C4—41no—
173C3×C4—41no—
174Dic3—82no—
175C2×C5#541411yescharacteristic
176C2×C5#542111yescharacteristic
177C2×C5#543011yescharacteristic
178D5—42no—
179C23—101no—
180Q8—31no—
181Q8—121no—
182Q8—121no—
183Q8—601no—
184Q8—601no—
185Q8#665511yescharacteristic, complemented
186D4—52no—
187D4—101no—
188C2×C4—31no—
189C2×C4—31no—
190C2×C4—121no—
191C2×C4—301no—
192C2×C4—601no—
193C2×C4—601no—
194C2×C4D5×S411yescharacteristic
195C8—121no—
196C8—302no—
197C8—601no—
198C2×C3—41no—
199C2×C3—41no—
200C2×C3—41no—
201C2×C3—802no—
202C5#7308711yescharacteristic, complemented, Sylow
203C22—102no—
204C22—201no—
205C22—201no—
206C22#9190411yescharacteristic
207C4—31no—
208C4—31no—
209C4—61no—
210C4—241no—
211C4—601no—
212C4#9189611yescharacteristic
213C4D5×GL(2,𝔽3)11yescharacteristic
214C3—41noSylow
215C2—202no—
216C2#1175158111yescharacteristic
217C2#1175161311yescharacteristic
218C2#1175162711yescharacteristic
subgroupG/H
centerC22#91904
commutator, cosocleC4×C5×SL(2,𝔽3)C22
Frattini, hypercenterC2×C4D5×S4
FittingC4×C5×Q8C2×D3
socleC22×C5GL(2,ℤ/4)
prime radicalSL(2,𝔽3)×D20C2
hypocenterC5×SL(2,𝔽3)SD16
LCSG ▷ C4×C5×SL(2,𝔽3) ▷ C2×C5×SL(2,𝔽3) ▷ C5×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C5×#5386C2semidirect
3. SL(2,𝔽3)⋊#168C2semidirect
24. #5386D5semidirect
48. C5×SL(2,𝔽3)SD16semidirect
96. C5×Q8#214semidirect
136. SL(2,𝔽3)#688semidirect
185. Q8#6655semidirect
202. C5#73087semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
SL(2,𝔽3)×D202
C5×#53862
SL(2,𝔽3)⋊#1682
#106019203
C2×#66554
#730875